How can I restrict the rendering of a parented object without removing the parent-child relationship?
The Outliner is designed to be able to display a 'heirarchy' of information, that is, data that belongs to other data, including parent/child relationships.

To expand the Outliner heirarchy press the '+' (which changes to a '-' when clicked) next to the parent object:



You now have access to the selection, render and viewport visibility properties of the child object. 

You can also press Numpad + or Numpad - to expand or collapse levels of the Outliner (while the cursor is in the Outliner) for multiple objects.

See the Outliner page in the Blender Manual for more information:

https://docs.blender.org/manual/en/dev/editors/outliner.html